66 F.M. Hemez and C.R. Farrar Fig. 6.6 Vibration testing of the ISS main truss. (a) NASA 8-bay truss testbed. (b) International Space Station main truss (courtesy: NASA [48]) Table 6.2 Correlation of the ISS main truss after calibration [48] Comparison of resonant frequencies Mode number Measured (Hz) Predicted (Hz) Error (%) Mode shape correlation (MAC) (%) 1 6.98 6.98 0.00 98.3 2 10.95 10.60 3.19 97.5 3 12.42 12.40 0.16 65.3 4 13.06 12.70 2.76 89.8 5 13.87 13.67 1.44 86.0 The versatility of ISBU has promoted application to many disciplines. One of them is Structural Health Monitoring (SHM), where model predictions augment physical measurements to locate, and assess the severity of, structural damage. Surveying the application of FE model calibration to SHM is beyond the scope of this discussion; a review can be found in [46]. One accomplishment, that we wish to highlight, is the work of Professor David Zimmerman, and his research team at the University of Houston, Texas. Their eigen-structure assignment technique [47, 48] was successful to calibrate models of the NASA International Space Station (ISS) main truss, illustrated in Fig. 6.6. Linear NASTRAN™ models with up to 66,000 DOFs were updated to achieve less than 3 % modal frequency error and at least 95 % mode shape correlation. These correlation requirements originate from recommended “best practices” [49]. Table 6.2 indicates a typical test-analysis agreement reached after FE model calibration for the ISS main truss. 6.4 A Brief Discussion of Challenges to Updating Methods The overview of FE model calibration is concluded by briefly discussing several issues that present challenges. The first three are practical issues that, unfortunately, need to be dealt with: ill-conditioning, spatial incompatibility and error localization. Our view is that these issues do not offer fundamental challenges to FE model updating. The next four topics are more foundational: information-theoretic limitations of test-analysis correlation, experimental variability, truncation error and the application of FE model updating to nonlinear dynamics. 6.4.1 Numerical Ill-Conditioning of Inverse Problems Inverse problems are, by definition, ill-posed. At best, it means that a parameter correction •p obtained by solving, for example, Eq. (6.27) or (6.31), is poor-quality due to contamination from numerical ill-conditioning. At worst, a calibration is ambiguous because an infinite number of solutions are available. Unless there are multiple solutions to choose from, ill-conditioning is not an issue that challenges FE model updating; it is an inconvenience to be dealt with. Several approaches are available to mitigate the effects of ill-conditioning, such as discussed in [16]. Regularization can be applied. The well-known Tikhonov regularization adds a “minimum perturbation” term to the cost function, the effect of which is to orient the search towards solutions that do not deviate too much from the nominal values of model parameters. Calibration methods formulated with the Bayesian statistical framework introduce a “prior” term whose effect is analogous to regularization. Lastly, good practices of numerical analysis can be implemented, such as filtering out the null space of the gradient matrix inverted.
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